The nonlinear Green's function method represents solutions of second-order equations α ∂ζ2w+N(w)=f\alpha\,\partial_\zeta^2 w + N(w)=f through a short-time expansion whose leading term convolves the source with G=θw0G=\theta w_0, where θ\theta is the Heaviside step function and w0w_0 solves the associated homogeneous Cauchy problem. Khurshudyan and Khurshudyan (2018) applied this scheme to wave equations whose nonlinearity depends on the absolute value of the unknown function, such as N(w)=∣w∣w3N(w)=|w|w^3 and N(w)=sin⁡∣w∣ wN(w)=\sin|w|\,w, validating it only numerically against the method of lines with fitted scale parameters (a0,s)(a_0,s). The open problem is to replace numerical validation with a certified approximation theory: convergence conditions on the nonlinearity, the source class, and the interval (including the finitely smooth case at w=0w=0); a priori computable error bounds, including how many terms achieve a prescribed tolerance; and a principled determination of the scale parameters (a0,s)(a_0,s), together with the dependence of the truncation error on the source class and on α\alpha. No later work addressing these items is known in the literature.

Background

For a linear inhomogeneous differential equation, the Green's function method writes the solution as a convolution of the source term with the impulse response of the operator. Superposition fails for nonlinear equations, but Frasca (Int. J. Mod. Phys. A 23 (2008) 299), in a strong-coupling analysis of quantum-field-theoretic equations, introduced a formal alternative for α ∂t2w+N(w)=f\alpha\,\partial_t^2 w + N(w) = f: a short-time expansion of the solution in which the leading term is the convolution of the source with a nonlinear Green's function G=θw0G=\theta w_0, where θ\theta is the Heaviside step function and w0w_0 is the general solution of the homogeneous equation α w0′′+N(w0)=0\alpha\,w_0''+N(w_0)=0 subject to suitable Cauchy data. Frasca and Khurshudyan later showed (Comput. Math. Methods 1 (2019) e1038) that this representation is exact whenever the nonlinearity satisfies a generalized homogeneity (multiplicativity) property θN(w)=N(θw)\theta N(w)=N(\theta w), and derived hierarchies of nonlinearities admitting it.

Khurshudyan and Khurshudyan (Int. J. Geom. Methods Mod. Phys. 15 (2018) 1850179) extended the scheme to wave equations whose restoring term depends on the absolute value of the unknown function, of the form N(w)=∣w∣ g(w)N(w)=|w|\,g(w) or N(w)=g1(∣w∣ g2(w))N(w)=g_1(|w|\,g_2(w)) — for instance N(w)=∣w∣w3N(w)=|w|w^3 and N(w)=sin⁡∣w∣ wN(w)=\sin|w|\,w, which arise in oscillation and damping models where the force law depends on the amplitude's sign. Because no closed-form w0w_0 was available for these nonlinearities, they computed w0w_0 by numerically integrating the homogeneous Cauchy problem, introduced scale parameters (a0,s)(a_0,s) fitted to minimize the deviation from a reference solution obtained by the method of lines (a standard numerical integrator that discretizes space and advances time as a system of ODEs), and checked second- and third-order truncations on representative numerical examples. Two structural obstacles distinguish this setting from earlier applications: the nonlinearity ∣w∣g(w)|w|g(w) is not differentiable at w=0w=0, so it is at best finitely smooth and the analyticity hypotheses of standard series-convergence arguments fail; and the scale parameters have no mathematical definition beyond error minimization, so nothing is known about how the truncation error depends on them, on the source function class, or on physical parameters such as α\alpha. The related Adomian decomposition method — an iterative series method expanding the nonlinearity in so-called Adomian polynomials — does possess a convergence theory for analytic nonlinearities via Banach-algebra and Volterra-type arguments (Abdelrazec and Pelinovsky, Numer. Methods Partial Differential Equations 27 (2011) 749), but it covers a different iteration scheme and its analyticity assumption excludes exactly the ∣w∣|w|-type nonlinearities at issue.

Problem Statement

For second-order wave equations α ∂ζ2w+N(w)=f\alpha\,\partial_\zeta^2 w + N(w) = f, where ζ\zeta is the independent variable (a rescaled time in the source paper), whose nonlinearity depends on the absolute value of the unknown function — specifically N(w)=∣w∣ g(w)N(w)=|w|\,g(w) or N(w)=g1(∣w∣ g2(w))N(w)=g_1(|w|\,g_2(w)) as treated in the source paper, with the source's examples N(w)=∣w∣w3N(w)=|w|w^3 and N(w)=sin⁡∣w∣ wN(w)=\sin|w|\,w — establish a rigorous approximation theory for the truncated short-time expansion (nonlinear Green's function) solution built on G=θw0G=\theta w_0. The expansion used in this literature (Frasca 2008; References 2 and 4, Eq. (3)) has the form

w(ζ)=a0∫0ζG(ζ−τ) f(τ) dτ+∑k=1∞ak∫0ζ(ζ−τ)k G(ζ−τ) f(τ) dτ,w(\zeta) = a_0\int_0^{\zeta} G(\zeta-\tau)\,f(\tau)\,d\tau + \sum_{k=1}^{\infty} a_k\int_0^{\zeta} (\zeta-\tau)^k\,G(\zeta-\tau)\,f(\tau)\,d\tau,

where G=θw0G=\theta w_0, w0w_0 solves the homogeneous Cauchy problem α w0′′+N(w0)=0\alpha\,w_0''+N(w_0)=0 with w0(0)=0w_0(0)=0, w0′(0)=sw_0'(0)=s, and the coefficients aka_k are determined by the Cauchy data w(k)(0)w^{(k)}(0); a KK-th order truncation keeps the terms with k≤Kk\le K. The objective is to convert the numerically benchmarked scheme into a certified one, comprising: (a) conditions on the nonlinearity, the source function class, and the interval under which the truncated expansion converges to the true solution, including the case of nonlinearities that are only finitely smooth at w=0w=0 — admissible regularity conditions on gg, g1g_1, g2g_2 are to be established rather than presupposed; (b) a priori computable error bounds, including a characterization of how many terms are needed for a prescribed tolerance on a given interval; and (c) a principled determination of the scale parameters (a0,s)(a_0,s), replacing their current definition as real parameters fitted to minimize the error against method-of-lines reference solutions, together with a description of how the truncation error depends on the source class and on physical parameters such as α\alpha.

The following further directions are related but are not required components of the objective: the behavior of the scheme on long intervals ζ≫1\zeta\gg1 and its extension to boundary-value problems where the short-time premise does not apply; and closed-form analytic expressions for the homogeneous solution w0w_0, and hence for GG, for the specific nonlinearity classes of the source paper.

References

  1. Asatur Zh. Khurshudyan and Martiros Khurshudyan, "Green's function solution of nonlinear wave equation depending on the absolute value of the unknown function", International Journal of Geometric Methods in Modern Physics 15 (2018) 1850179, DOI 10.1142/S0219887818501797, https://doi.org/10.1142/S0219887818501797
  2. Marco Frasca and Asatur Zh. Khurshudyan, "A general representation for the Green's function of second order nonlinear differential equations", Computational and Mathematical Methods 1 (2019) e1038, DOI 10.1002/cmm4.1038, arXiv:1805.10495, https://arxiv.org/abs/1805.10495
  3. Marco Frasca and Asatur Zh. Khurshudyan, "Green's functions for higher order nonlinear equations", International Journal of Modern Physics C 29 (2018) 1850104, DOI 10.1142/S0129183118501048, arXiv:1806.11313, https://arxiv.org/abs/1806.11313
  4. Marco Frasca and Asatur Khurshudyan, "General Representation of Nonlinear Green's Function for Second Order Differential Equations Nonlinear in the First Derivative" (2018), arXiv:1806.00274, https://arxiv.org/abs/1806.00274
  5. Asatur Zh. Khurshudyan, "An identity for the Heaviside function and its application in representation of nonlinear Green's function", Computational and Applied Mathematics 39 (2020), article 32, DOI 10.1007/s40314-019-1011-5, https://doi.org/10.1007/s40314-019-1011-5
  6. Ara S. Avetisyan and Asatur Zh. Khurshudyan, "Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green's function approach", ZAMM — Journal of Applied Mathematics and Mechanics 98 (2018) e201800122, pages 1992–2009, DOI 10.1002/zamm.201800122, https://doi.org/10.1002/zamm.201800122
  7. Marco Frasca, "Green functions and nonlinear systems: Short time expansion", International Journal of Modern Physics A 23 (2008) 299–308, DOI 10.1142/S0217751X08038160, arXiv:0704.1568, https://doi.org/10.1142/S0217751X08038160
  8. Ahmed Abdelrazec and Dmitry Pelinovsky, "Convergence of the Adomian decomposition method for initial-value problems", Numerical Methods for Partial Differential Equations 27 (2011) 749–766, DOI 10.1002/num.20549, https://doi.org/10.1002/num.20549